cp's OEIS Frontend

Tentatively, as of Dec 2012, the likely value of a(6) is 20968. A noteworthy fact, perhaps, is that were this sequence to limit itself to non-titanic primes (ones under 10^999), then it would look the same to the point shown and have the stated tentative value for a(6) as its a(5), despite there being a number of smaller values eventually reaching a 5th prime. - James G. Merickel, Dec 06 2012

a(5)=8338 has not been determined with complete certainty, but is likely correct (See A232657). a(6)=20968 has fairly convincing support, but even finding a good upper bound for a(7) is hard. - James G. Merickel, Jun 14 2014

Excluding the prime a(n) itself, this sequence concerns stringing together (in decimal notation) values a(n), a(n)-1, a(n)-2, ... to obtain n primes as quickly as possible. A remark (from Jens Kruse Andersen, responsible also for a(4)) that beginning at a prime is more common than what had been done at A152396 is responsible for initiating investigation of this sequence. To be specific, this sequence concerns first instances of numbers resulting in simultaneous primality of concatenations of k numbers where k runs through the first n+1 (the number by itself is prime) odd values not divisible by 3 or ending with digit 9 in decimal (the first condition concerns simple divisibility by 3, while the second results from the small values placing a specific limitation modulo 5). No splitting into two distinct sequences arises depending upon whether or not it is specified that each prime occurs with as few concatenations as possible. This contrasts with what occurs where the number itself need not be prime (and may not be counted). See the comments to A152396 for the complications that arise in that case. - James G. Merickel, Feb 26 2014

Existence of solutions appears provable using Chebotarev's Theorem applied to the rational primes, after first applying the Chinese Remainder Theorem to fixed-length collections. - James G. Merickel, Mar 11 2014

The very simplest example of the type of prime concerned is 43, but 100 (see Example) is the first value to produce 4 primes. 1000 is the first to do so in the fastest way possible. Proper concatenation first excludes 73 from this list, as the stem--initial--value in concatenation is itself excluded from consideration.

8338 is the first value to produce 5 primes, with terminal values 8323, 8293, 8079, 7561 and 6519.

Analogous to A152396 with increments replacing decrements.